One of the most famous unsolved problems in mathematics is a conjecture made by Christian Goldbach in a letter to Leonhard Euler in 1742. The conjecture made in this letter is now known as Goldbach's Conjecture. The conjecture is as follows:
Every even integer greater than 2 can be expressed as the sum of two (not necessarily distinct) prime numbers.
Currently, it is not known if this conjecture is true or false, although most mathematicians believe it to be true.
(a) If Goldbach's conjecture is false, describe one way to prove it.
(b) Prove the following: If there exists an odd integer greater than 5 that is not the sum of three prime numbers, then Goldbach's Conjecture is false.